Definition (under construction)

In the general case we should distinguish left and right bialgebroids, see bialgebroid.

In one of the versions, a general Hopf algebroid is defined as a pair of a left algebroid and right algebroid together with a linear map from left to right bialgebroid taking the role of an antipode (…).

Noncommutative Hopf algebroids

There are several generalizations to the noncommutative case. A difficult part is to work over the noncommutative base (i.e., the object of objects is noncommutative). The definition of a bialgebroid is not that difficult and there is even a very old definition due Takeuchi. To add an antipode is nontrivial. A definition of Lu from mid 1990s is rather nonselfdual unlike the case of Hopf algebras. So a better solution is to abandon the idea of an antipode and have some replacement for it. There are two approaches, one due to Day and Street, and another due Gabi Böhm, using pairs of a left and right bialgebroid. Gabi later showed that the two definitions are in fact equivalent.

Examples

Scalar extension Hopf algebroids

Given a Hopf algebraBB and a braided-commutative algebra AA in the category of Yetter-Drinfeld modules over BB, the smash product algebra B♯AB\sharp A is the total algebra of a Hopf algebroid over AA. This is a noncommutative generalization (of formal dual of) an action groupoid.

The modern concept over the noncommutative base has been discovered by several different people in several different formalisms. Some of the differences are merely cosmetic, but there are at least two main concepts, depending on the underlying concept of ‘bialgebroid’.

In this they start by taking an algebroid to be an “algebra with several objects” in the sense of a kk-linear category AA: that is, a VV-enriched category with V=VectkV = Vect_k. The 2-category VCatV Cat of kk-linear categories, functors and natural transformations is monoidal (where the tensor product of VV-categories is defined by cartesian product on object sets and tensor product on hom-spaces). So, they define a bialgebroid to be a comonoid in VCatV Cat. Because the tensor product is cartesian product on object sets, the comultiplication in such a bialgebroid is forced to be the diagonal on objects. Thus, their notion of bialgebroid amounts to a kk-linear category AA equipped with linear maps

A(a,b)→A(a,b)⊗A(a,b) A(a,b) \to A(a,b) \otimes A(a,b)

satisfying coassociativity, a version of the usual bialgebra axiom, and so on. On page 142 of the above reference they define an antipode on a bialgebroid AA to be a kk-linear functor S:A→AopS: A \to A^{op} together with a natural isomorphism

This starts with a different concept of bialgebroid, which is discussed here on the nLab. Namely: any kk-algebra RR gives a pseudomonoid Re=Rop⊗RR^e = R^{op} \otimes R in the bicategory ModkMod_k of k-algebras, bimodules, and bimodule homomorphisms, and a bialgebroid is then an opmonoidal monad AA on ReR^e. When the fusion (or Galois) operator for this opmonoidal monad is invertible, we say that AA is a Hopf algebroid. In G. Böhm’s work this definition is stated in a less compressed, more down-to-earth way.